After a creature is countered by a spell such as Counterspell can it regenerate. I was trying to counter my friend's Mortivore, but he said it could regenerate from that. So I need help trying to figure if he is lying or not.
No, a creature cannot use Regenerate to avoid being countered.
Counterspell uses the Counter keyword action, which removes a spell from the stack. This means the spell never resolves and does not have any effect. In the case of a creature spell, the effect is to put a creature on the battlefield; if it is countered, the creature is never created.
The abilities on a creature card (including regeneration) cannot be used until the card has finished being cast and become a creature on the battlefield*. Therefore, Regenerate cannot be used while Mortivore is on the stack.
Since Counterspell gets rid of Mortivore while it is still on the stack, your friend cannot use Mortivore's regeneration ability to avoid your Counterspell.
Note: Even if you could use a regeneration ability from the stack, it still wouldn't apply here. Regeneration only applies when a permanent would be Destroyed, which is different than being countered.
*Except in certain exceptional cases, none of which apply here.
There's no possible interaction between Counterspell and regeneration.
You can't use Counterspell on a creature (a creature card or token on the battlefield), only on a creature spell (a creature card or copy thereof on the stack).
Yet Mortivore's activated ability can only be activated when it's a permanent, not a spell. This is the case for most activated abilities. One can only activate an abilities of object in other zones if they are in the only zone in which it makes sense to activate the ability. (e.g. Cycling on cards such as Akroma's Blessing can only be activated from one's hand since only cards in hand can be discarded.)
In fact, regeneration can only be given to permanents (cards or tokens on the battlefield), not spells (cards or copies thereof on the stack).
109.2. If a spell or ability uses a description of an object that includes a card type or subtype, but doesn’t include the word “card,” “spell,” “source,” or “scheme,” it means a permanent of that card type or subtype on the battlefield.
110.1. A permanent is a card or token on the battlefield. [...]
112.6. Abilities of an instant or sorcery spell usually function only while that object is on the stack. Abilities of all other objects usually function only while that object is on the battlefield. The exceptions are as follows: [No exceptions apply to Mortivore's activated ability.]
You cannot regenerate a countered creature.
As a general rule, unless an ability only makes sense when applied in another zone (e.g. "return this card from graveyard to hand" clearly must work in the graveyard and "cannot be countered" must function on the stack) then that ability only applies while the card is a permanent on the battlefield.
Regenerate is one such ability. Countering a creature spell prevents it from entering the battlefield, so it never had an opportunity to be regenerated.
Furthermore, Regenerate prevents the destruction of a permanent, but countering a spell doesn't destroy it. So even if you could use regenerate while it was on the stack the ability wouldn't do anything.
Regenerate means "The next time this creature would be destroyed this turn, instead tap it, remove all damage done to it, and if it is in combat, remove it from combat." Countering does not destroy, partially because "destroy", like many words in magic, is a word with a very precise meaning and countering doesn't say that it destroys, and partially because a spell is not on the battlefield, and destroy is exclusively relevant to things on the battlefield. So regeneration is not applicable, the same way you can counter a creature with indestructible.