No - it's still an attack
Per the find familiar spell description:
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can't attack, but it can take other actions as normal.
If it cannot attack as its first action, it cannot attack on a haste-induced action.
Now, familiars of warlocks with the Pact of the Chain feature are allowed to attack:
when you take the Attack action, you can forgo one of your own attacks to allow your familiar to use its reaction to make one attack of its own.
But even if you were hasted and got two attacks per round, the familiar would still only get one attack, as it only has one reaction per round.