Only if it becomes a separate adventurer and takes actual class levels.
One of the requirements for Animal Ally is:
character level 4th
Character level is defined in the glossary as:
The total level of the character, which is the sum of all class levels held by that character.
And classes are defined as:
Classes represent chosen professions taken by characters and some other creatures. Classes give a host of bonuses and allow characters to take actions that they otherwise could not, such as casting spells or changing shape. As a creature gains levels in a given class, it gains new, more powerful abilities. Most PCs gain levels in the core classes or prestige classes, since these are the most powerful. Most NPCs gain levels in NPC Classes, which are less powerful.
This does not include racial hit dice, which is the only thing a companion gets. So unless you upgrade your companion to a separate character and then make it earn enough XP to take 4 class levels, it won't qualify for Animal Ally.