Yes, the firefly larvae that people call glowworms do turn into fireflies, because fireflies are beetles that go through complete metamorphosis.
According to the Xerces Society, every firefly passes through four stages: egg, larva, pupa and adult. The whole cycle can take anywhere from a couple of months to two or three years or more, and most of that time is spent as a larva hunting snails, slugs and other soft-bodied prey in moist soil. The larvae of all firefly species glow, which is why the young are so often called glowworms. Most fireflies pupate underground or in rotting logs, and adults typically emerge in late spring or early summer. Not every adult lights up, though: daytime dark fireflies make no light as adults. In the group known as glow-worm fireflies, the adult females keep a larva-like body and cannot fly because their wings are short or missing, while the males look like typical fireflies. So a glowing, grub-like insect in the grass may already be an adult female rather than a larva waiting to change. More than 2,200 firefly species have been described, about 165 of them in the United States and Canada.
Source: Xerces Society, About Fireflies
Image: Firefly larva (Pyractomena species), Judy Gallagher, CC BY 2.0, via Wikimedia Commons.