Martial did write one epigram on the subject (VI.57):

Mentiris fictos unguento, Phoebe, capillos
Et tegitur pictis sordida calva comis.
Tonsorem capiti non est adhibere necesse:
Radere te melius spongia, Phoebe, potest.

translation
You manufacture, with the aid of unguents, a false head of hair,
and your bald and dirty skull is covered with dyed locks.
There is no need to have a hairdresser for your head.
A sponge, Phoebus, would do the business better.

This is, as far as I know, the only explicit description of bald men painting on hair. Hair dye was used frequently by balding men to give hair a fuller appearance.

As Martial notes, it's an oily dye, and it clings to skin as much as to hair. So given that this is the only reference to painting on hair, and that hair dye is well attested, it seems more likely that he's talking about an old man with wispy hair, who tries to make it appear fuller with the addition of dye. But he is generous with the dye, thinking it will fill in the gaps for him as well. Sort of the equivalent of an extreme combover.

But given the silence on the subject, it seems unlikely that this should be understood as a completely bald man painting a hairdo on his bald head. Just a balding guy getting a bit aggressive with the hair dye. That being said, the language is explicit that he is in fact painting hairs on his head.

There's plenty out there on Roman hairstyles, and even the social history of baldness. The usual cover-ups, combovers, and wigs. But if painting a full head of hair on a bald head were a fashion, there would almost certainly be more than this single epigram (absence of evidence, I know. But given the frequency with which authors liked to talk about style, the absence is deafening). It's more likely a slight exaggeration about the overuse of dye to stain the scalp as well as hair.

Edit: Someone beat me to it! My post doesn't add much, but here you go:

This is surely the Martial epigram you're referencing (Book 6, epigram 57):

Mentiris fictos unguento, Phoebe, capillos

et tegitur pictis sordida calua comis.

Tonsorem capiti non est adhibere necesse:

radere te melius spongea, Phoebe, potest.

Translation:

You make fake hair with gel, Pheobus—

Your filthy scalp is covered with drawn-in hair.

You don’t need to summon a barber for that head:

A sponge can give you a better shave, Phoebus.

The operative word concerning the "drawn-in hair" is pictis, which, according to Lewis and Short's entry is the same sort of word you would use if you were painting with a brush or pencil. (Interestingly some authors used it to mean "tattooed" but only in reference to non-Roman customs, so I doubt that's what Phoebus is doing here. Besides, if it comes off with a sponge it is hardly tattooed.)

Honestly, I can't find a single source for this other than the Martial epigram, and in my experience you need to take Martial with a grain of salt just because he's satire. It could be possible that some Romans painted hair on their heads, but it's equally possible that he's just making fun of some guy whose thin, short, greasy hair looks like it could be painted on. It's like a future historian seeing someone compare Donald Trump's hair to a fox or something, and saying that 21st-century Americans used animal pelts as wigs.

So Martial is inconclusive in my opinion, but it's certainly possible! In any case the practice wasn't widespread.