Can we restrict today's date to accept reservation from booking engine?

Yes, We can restrict today's date to accept a reservation from booking engine. 

In Channel manager, from Settings >> Booking page tab you can set. If you set 1 here, which means your booking will start from the next day. The system will not allow you to take the booking on the same day. 

Clarification: Prior to one-day booking need to be done. Let’s say, I want my booking on 4^th  of Feb and the number I have set is 0. Which means I can book my booking on 4^th  Feb. system will allow me this. But, If I change the number 0 to 1 from reservation panel then it won’t allow me the same day booking. Please, check below screenshot.

The above screenshot for the Number 0, which we have configured. Now, let’s change the number from 0 to 1 and check it.

As you can see on above two screenshots when you set number 0 at -Enter no of days which reservations can be accepted. So, on the same day, you can take the booking. But, if you change it to 1, it will not take the booking for the same day. Booking will start from the next day.