If the column in question is NOT NULL, both of your queries are equivalent. When group_id contains null values,
select count(*)
will count all rows, whereas
select count(group_id)
will only count the rows where group_id is not null.
Also, some database systems, like MySQL employ an optimization when you ask for count(*) which makes such queries a bit faster than the specific one.
Personally, when just counting, I’m doing count(*) to be on the safe side with the nulls.