Yes, if the caster makes it sound "reasonable"
The relevant part of the description of the suggestion spell states (emphasis mine):
You suggest a course of activity (limited to a sentence or two) and magically influence a creature you can see within range that can hear and understand you. Creatures that can't be charmed are immune to this effect. The suggestion must be worded in such a manner as to make the course of action sound reasonable. Asking the creature to stab itself, throw itself onto a spear, immolate itself, or do some other obviously harmful act ends the spell.
As written, it says the suggested course of action must "sound reasonable", and that asking the creature to do an obviously harmful act ends the spell. I'll address those issues in reverse order, since the latter is more straightforward.
"Obviously harmful act" refers to direct physical harm
The last sentence of the first paragraph of the spell description says that asking the creature to perform an "obviously harmful act" ends the spell. On its own, it may seem that this phrase is very vague as to what it means.
However, the examples listed earlier in the same sentence suggest that the restriction is limited to direct physical harm - not hurt feelings, or the risk of retaliation by someone that thought you were their friend, or anything more abstract. Rules designer Jeremy Crawford unofficially confirms this interpretation in this 2016 tweet:
Suggestion 'obviously harmful' meaning immediately life threatening? "tell us where your secret base is"?"surrender"?
It means physical harm.
Considering that all the examples given in the spell description of "obviously harmful acts" refer to the creature attacking itself, throwing itself onto someone else's weapon, or into an obviously lethal environmental hazard, this seems like a relatively straightforward interpretation of what it means by "obviously harmful".
Attacking an ally may be reasonable, depending on the suggestion's wording
This closed Q&A addresses the issue of what counts as "reasonable" for the suggestion spell, but we can also look at a later part of the spell description for a suggested course of action and triggering condition that ostensibly works for the spell (emphasis mine):
You can also specify conditions that will trigger a special activity during the duration. For example, you might suggest that a knight give her warhorse to the first beggar she meets. If the condition isn't met before the spell expires, the activity isn't performed.
As we can see from this, what counts as "reasonable" may be fairly broad/open-ended. As long as you can justify the suggested course of action in a reasonable-sounding way within a sentence or two, there's a lot you can convince the target of suggestion to do.
Jeremy Crawford also provides some unofficial guidance on some ways in which the suggestion spell might be used in combat in this June 2016 tweet:
can you give me some examples of the suggestion spell in combat? Thank you sir.
Suggestion examples: "Flee! A dragon comes." "Don't attack; I intend no harm." "Your sword is cursed. Drop it."
In these cases, none of the examples involve telling the enemy to attack their ally - but even so, not attacking the caster/the caster's allies or dropping their weapon in combat is still pretty powerful, so suggestion's capabilities should not be underestimated.
In addition, fellow D&D designers Chris Perkins and Mike Mearls both independently responded (unofficially) to this June 2017 tweet that asks whether the suggestion spell can be used to make the target attack another creature:
The spell suggestion. Can it cause target to attack another creature, or is it no physical harm to ANY target?
Perkins: It can, but if the spell's target thinks the suggested action is likely to cause it harm, the spell could fail.
Mearls: Only in specific situations - would have to play upon existing rivalry or anger. Ex - Guy you suspect is a traitor really is.
Note that Perkins' tweet here seems to interpret "obviously harmful act" a bit more liberally than Crawford's does above; he says thinking the action's "likely to cause it harm" may be enough to meet the criterion. However, this isn't quite backed up by the spell description. Nevertheless, Perkins does agree with the base idea: suggestion can tell you to attack a target.
Mearls' response is slightly more detailed. He says the caster has to play on the target's "existing rivalry or anger" against the other creature to convince the target to attack it. This isn't quite what suggestion says; the spell description merely says the suggested course of activity has to be worded so it sounds "reasonable".
Mearls' approach seems a bit too restrictive, as it doesn't allow for situations where the sentence or two you speak as part of the suggestion spell themselves set up the "reasonability" of the suggested course of action (look at the Crawford tweet quoted above for examples). The premise of the suggestion need not be true, only "reasonable".
Remember that suggestion doesn't compel a belief; it compels a course of action if the course of action is made to sound reasonable. (See the Q&As "Can the Suggestion and Mass Suggestion spells alter feelings?" and "Do the ideas that support the Suggestion spell persist after the end of the spell?".)
Note that per the spell description, the "reasonable" criterion is only relevant at the time of initial casting; once this initial requirement is met, the target (if it fails a Wisdom save) will continue to obey this course of action until it is completed or until the spell ends. (Crawford unofficially confirms this ruling in this December 2017 tweet.)
If the caster does manage to word the suggestion convincingly enough in just the sentence or two specified, and the target fails its Wisdom save, then it would pursue the course of action to the best of its ability.
That said, this Q&A about who decides whether a suggestion cast by an NPC on a player character is "reasonable" is very relevant and worth keeping in mind. As Rubiksmoose's answer there says: "Ideally it should be agreed on, but DM has the final call".
Especially when using a mind-control spell that inherently takes away player agency, it's important to find a way to resolve the situation that everyone's willing to accept (even if they're not happy about their PC being mind-controlled...); there needs to be a willingness by both player and DM to trust that the other is behaving/arguing fairly.
Addressing the example from The Rise of Tiamat specifically
I wanted to address the example from the adventure The Rise of Tiamat separately.
As quoted in the question, the end of the description of
area 15 ("Bridge") of the Tomb of Diderius/Ss'tck'al (on p. 46 of the original Rise of Tiamat book, or p. 132 of the combined Tyranny of Dragons rerelease)
says (emphasis mine):
[...] The [yuan-ti malisons (type 1)] also use suggestion spells to tell characters that a fellow party member has been replaced by a yuan-ti spy and should be attacked.
Saying that "a fellow party member has been replaced by a yuan-ti spy" certainly words things in a way that makes it reasonable to treat the party member with suspicion and hostility, and perhaps even attack them... But again, suggestion doesn't implant beliefs in the target; it just forces them to take a course of action if it's worded to "sound reasonable".
In the context of the description of the room it appears in, the suggestion as written is unlikely to come off as reasonable. The spell is expected to be cast while the party is fighting not only the yuan-ti malisons (type 1) themselves, but also several lizardfolk. In addition, unless the supposed "yuan-ti spy" has been separated from the rest of the party at some recent point in time, the party's fought alongside that character (including potentially against yuan-ti).
Unless the party member who's claimed to be a spy by the suggestion is already being suspicious or has betrayed/attacked the party before, it will seem like the yuan-ti are effectively saying "your long-time friend who's attacking us and not you is actually a spy working for us, so attack him". Not only is the premise difficult to consider reasonable, but the suggested course of action ("attack him") also doesn't seem like the most reasonable response even if that premise were accepted. After all, a "spy" whose cover is blown may not be the highest-priority target.
It's worth keeping in mind that the Tyranny of Dragons adventures, Hoard of the Dragon Queen and The Rise of Tiamat, were written not by Wizards of the Coast, but by a third party, Kobold Press. The former book was released on the same day as the PHB, and the latter was released between the MM and the DMG. Dungeon Masters and players have noticed several issues with the adventures as a result of this process, including poorly balanced encounters and some rules inaccuracies; the way suggestion is implied to work here may be one such error.