Yes, better yet, they had actual toothpicks, called dentiscalpia.

Martial describes Roman toothpicks in Ep. 14.22, entitled Dentiscalpium:

Lentiscum melius: sed si tibi frondea cuspis
Defuerit, dentes pinna levare potest.

A piece of Lentisc wood is best;
but if that is unattainable, a quill may relieve your teeth.

In another epigram (7.53) he jokes about how dentiscalpia are cheap and worthless gifts.

So the Romans had toothpicks just like us. They also had dentifricium, a powder for cleaning teeth, dentilegus, a person who collects teeth, and a dentharpaga, an instrument for removing teeth...