Minions have their own turns and can be Quickened.

Your GM is probably interpreting the minion rules to mean that minions don't get their own turns.

A creature with this trait can use only 2 actions per turn, doesn't have reactions, and can't act when it's not your turn. Your minion acts on your turn in combat, once per turn, when you spend an action to issue it commands.

If given no commands, minions use no actions except to defend themselves or to escape obvious harm.

Suppose minions didn't get their own turns. Because the Haste spell gives the Quickened condition, a creature could only benefit from this condition if they had a turn to start.

You gain 1 additional action at the start of your turn each round.

Because quickened has its effect at the start of your turn, you don’t immediately gain actions if you become quickened during your turn.

If a creature doesn't have its own turn, how would that affect other game mechanics? All minions would be unaffected by conditions like Dying, Slowed, and Stunned. They would be immune to Persistent Damage, and they don't incur Multiple Attack Penalty.

Does that seem reasonable? Is that what your GM wants?

There are also mechanics such as Spark of Independence that would not make any sense with this ruling.

The minion is quickened for 1 round and is then temporarily immune for 10 minutes. It can use this extra action to Step, Stride, or Strike.

Regardless of Paizo's notorious editing habits, the fact that they published such am ability means that some of the game designers think that minions get turns. So what does this all mean?

A reasonable interpretation is that minions do have turns, just like all other creatures participating in combat encounters. But by default, a minion has zero actions on its turn. A minion's turn starts when its controller's turn starts, and a minion's turn ends when its controller's turn ends.

Therefore, a minion can be Quickened, Slowed, and so on. If you cast Haste on your own minion, then they benefit from Quickened at the start of their next turn, which happens at the start of your next turn. And even if it gets extra actions, a minion cannot use actions unless commanded.

Related: an answer to "How many actions can a summoned zombie use?"

Minions have their own turns and can be Quickened.

Your GM is probably interpreting the minion rules to mean that minions don't get their own turns.

A creature with this trait can use only 2 actions per turn, doesn't have reactions, and can't act when it's not your turn. Your minion acts on your turn in combat, once per turn, when you spend an action to issue it commands.

If given no commands, minions use no actions except to defend themselves or to escape obvious harm.

This trait doesn't say that minions don't get turns. It just says their turns work a bit differently.

The Haste spell gives the Quickened condition, so a creature can only benefit from this condition if they have a turn to start.

You gain 1 additional action at the start of your turn each round.

Because quickened has its effect at the start of your turn, you don’t immediately gain actions if you become quickened during your turn.

If your GM is right, what else would happen if minions don't have their own turns? All minions would be unaffected by conditions like Dying, Slowed, and Stunned. They would be immune to Persistent Damage, and they don't incur Multiple Attack Penalty. None of this seems reasonable.

There are also mechanics such as Spark of Independence that indicate minions are affected by these conditions:

The minion is quickened for 1 round and is then temporarily immune for 10 minutes. It can use this extra action to Step, Stride, or Strike.

Although this feature is from an expansion book, it is clear evidence that minions can be Quickened.

A reasonable interpretation is that minions do have turns, just like all other creatures participating in combat encounters. But by default, a minion has zero actions on its turn. A minion's turn starts when its controller's turn starts, and a minion's turn ends when its controller's turn ends.

Therefore, a minion can be Quickened, Slowed, and so on. If you cast Haste on your own minion, then they are Quickened, and they gain the extra action at the start of their next turn, which happens at the start of your next turn. And even if it gets extra actions, a minion cannot use actions unless commanded on that turn.

Related: an answer to "How many actions can a summoned zombie use?"