Yes, touch spells work in adjacent squares
If you are adjacent to another creature, you are 5' away, as NautArch explains here.
In the PHB (pg. 202) under Range, it says this about Touch range:
Some spells can target only a creature (including you) that you can touch.
This doesn't really help much, but we can infer that to perform a touch spell on that creature, you do not have to occupy their space, since you can't end your movement there, so touch spells must effectively have a range of 5ft if they are to make any sense (although they don't actually have a range of 5ft or any feet; it's simply useful to think of it as such for the purposes of a grid).
If I include the quote from linksassin's answer from the PHB (pg. 195), spells with a range of touch that require an attack roll of any kind are melee spell attacks (i.e. the spell's description will say "make a melee spell attack") and are therefore subject to this quote:
Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack. Certain creatures (typically those larger than Medium) have melee attacks with a greater reach than 5 feet, as noted in their descriptions.
Although not all touch spells require attack rolls (since not all of them are attacks, such as cure wounds), we can at least infer from those that do that touch spells can work in adjacent squares nonetheless.
So long as you can touch them without moving out of your 5ft square (i.e. if they are adjacent to you), you can target the creature with a spell with a range of Touch.