Yes, the rules for this are in the Monster Manual
Suggestion just requires a suggestion that sounds reasonable within a sentence or two. You could suggest the following as a very reasonable suggestion:
"I'll make you a deal demon. You tell me your true name so I can protect myself from you on the prime material plane, and I'll release you to wreak havoc on anything you please, excluding me."
In the MM pg. 53, there's an excerpt directly related to this, emphasis mine:
DEMON TRUE NAMES
Though demons all have common names, every demon lord
and every demon of type 1 through 6 has a true name that
it keeps secret. A demon can be forced to disclose its true
name if charmed, and ancient scrolls and tomes are said to
exist that list the true names of the most powerful demons.
A mortal who learns a demon's true name can use
powerful summoning magic to call the demon from the
Abyss and exercise some measure of control over it.
However, most demons brought to the Material Plane in this
manner do everything in their power to wreak havoc or sow
discord and strife.
However this also highlights that you would only have some measure of control, not permanent power. The follow up sentence in the description provided highlights that the demon will actively fight you on this, making Planar Binding a requirement to force something into servitude.
On a personal side note: This is why I use Devil's instead of Demons. Devil's don't like service, but they expect to be rewarded and will willingly follow commands when they are issued. Source is MM pg. 66, under the section Obedience and Ambition.