They certainly did, and very frequently as well. Directly naming your child after a god (e.g. calling them Zeus, for example) was not done, though there are a few rare exceptions after the first century BC. But there were a number of names that were created by attaching a suffix to a god's name, and the three most common Greek names were all of this sort (called theophoric).

These three were Dionysius (from which we get Dennis), Apollonius, and Demetrius. All of them are formed by attaching the -ios suffix (or -ia if you were naming a girl or city) to the name of a god (Dionysus, Apollo, Demeter). This suffix was a grammatical suffix that indicated a slightly vague concept of 'belonging to' (e.g. Alexandria, named because it 'belonged to' Alexander). So anyone named in this way had a name meaning 'belonging to Dionysus/Apollo/Demeter).

Another popular suffix was -doros, derived from the Greek word doron ('gift'). Popular names of this sort included Apollodorus, Isidorus, Diodorus; this meant 'gift of Apollo/Isis/Zeus'.

Another was -genes, from the verb gignomai ('to come into being'). This had the meaning 'born from X'. The most popular name of this sort was Diogenes ('born from Zeus'); this is also found as a generic adjective in early Greek literature such as the Iliad, applied as an epithet to kings or other heroes, and indicating their connection to the king and greatest of the gods. Similar names are Hermogenes ('born from Hermes') and Theogenes ('born from a god').

Finally there are names formed with -kles, from the word kleos ('glory'). The most famous name of this kind is obviously Herakles ('glory of Hera'), but we also find Diokles ('glory of Zeus') and Dionysikles ('glory of Dionysus').

Now, not all gods had people named after them: Ares generally didn't, because he wasn't a particularly great god, and nor did Hades (naming your child after the god of the underworld was very bad vibes). And despite these suffixes theoretically being interchangeable, in practice they often weren't, or at least not in equal number: Apollokles and Apollogenes were very, very rare; so was Hermios. So yes, the Greeks did name their children after their gods, but not all gods, and although there was a lot of flexibility in name formation, it was not absolute. If you're interested in ancient Greek names more generally, there is a project called the Lexicon of Greek Personal Names (LGPN) that aims to collect and catalogue every name (that we have evidence for). Its database is quite easy to use, and provides lots of information, such as where and when a name is attested: https://www.lgpn.ox.ac.uk/