No — gophers are not the same animal as groundhogs, but groundhog and woodchuck are two common names for the same species, Marmota monax.
University of Missouri Extension IPM describes them separately: the plains pocket gopher (Geomys bursarius) is a smaller burrowing rodent with cheek pouches used to carry food, while groundhogs or woodchucks (Marmota monax) are medium-sized rodents (about 16–27 inches long) that hibernate in burrows and leave larger open entrance holes. Pocket gophers belong to the family Geomyidae; groundhogs are marmots in the squirrel family (Sciuridae).
National Geographic likewise notes that groundhogs are also known as woodchucks and are marmots closely related to squirrels — not pocket gophers.
Source: University of Missouri Extension IPM — Rodentia Get Off My Property (gophers vs. groundhogs/woodchucks); also National Geographic — 10 facts about groundhogs.