If you can safely make (firstName, lastName) the PRIMARY KEY or at least put a UNIQUE key on them, then you could do this:
INSERT INTO logins (firstName, lastName, logins) VALUES ('Steve', 'Smith', 1)
ON DUPLICATE KEY UPDATE logins = logins + 1;
If you can't do that, then you'd have to fetch whatever that primary key is first, so I don't think you could achieve what you want in one query.