The ability triggers, but can't use these lands to activate Borborygmos Enraged's ability
The ability triggers when Borborygmos Enraged deals combat damage, even if at the same time it's dealt lethal damage itself. The ability is put on the stack, but state-based actions see lethal damage marked on it and destroy it, causing it to go into your graveyard, before you get priority and could activate the ability.
If nothing else happens, the ability will go on the stack, resolve, and possibly reveal an amount of lands you may put into your hand.
Relevant Rules
510.2. Second, all combat damage that's been assigned is dealt simultaneously. (...)
510.3. Third, the active player gets priority. (...)
510.3a. Any abilities that triggered on damage being dealt or while state-based actions are performed afterward are put onto the stack before the active player gets priority (...)
704.3. Whenever a player would get priority (...), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. (...) Once no more state-based actions have been performed as the result of a check and no triggered abilities are waiting to be put on the stack, the appropriate player gets priority. (...)