No. The Borderlands series (1, 2, The Pre-Sequel, 3) supports local split-screen co-op (2 players) only in offline mode. You cannot take a split-screen party into online matchmaking or join an online session together from one console.
Note: Borderlands 3 added 3/4-player split-screen on next-gen consoles (PS5/Xbox Series X), but still offline only.