This will depend on your DM

According to the text for a readied action:

Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn so that you can act later in the round using your reaction. First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.

So, you can ready an action during a surprise round, however in order to use it you'll need to give your DM the perceivable circumstance that you're interested in. Some DMs may consider 'the enemy starts their turn' as perceivable, but some may not. It is definitely a 'meta' approach (turns aren't real things, they're abstractions), and many DMs would not let you do that.

Also, consider the fact that the text later says:

When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.

So whatever trigger you set has to occur first, then you can act. So if you say something like 'the enemy begins their turn', and your DM accepts that, the DM is well within their rights to determine what the 'beginning' of the turn includes. That may allow the target to attack back, slink away or even take the dodge action at the beginning of their turn, which would make it harder for you to hit your action (or even potentially lose your action altogether).