If I activate the Lightsail 2TB plan in the middle of the month, can I utilize the entire 2TB quota within that same month?

Hello.

First of all, the Lightsail instance is charged even if it is stopped.
https://aws.amazon.com/lightsail/faq/?nc1=h_ls

Q: When am I getting charged for a plan?
Your Lightsail instances are charged only when they're in the running or stopped state. If you delete your Lightsail instance before the end of the month, we only charge you a prorated cost, based on the total number of hours that you used your Lightsail instance. For example, if you use the least expensive Lightsail plan for 100 hours in a month, you will be charged 47 cents (100*0.0047).

Additionally, if you exceed the amount of communication included in your Lightsial instance plan, you will be billed for the excess amount.

Q: What does data transfer cost?
Your plan includes a free data transfer allowance. Both data transfer in and data transfer out of your instance count toward your data transfer allowance. For Lightsail content delivery network (CDN) distributions, only data transfer out of your distribution counts toward your allowance.
If you exceed your data transfer allowance, you will only get charged for data transfer OUT from a Lightsail instance to the Internet or to AWS resources using the public IP address of the instance. Both data transfer IN to Lightsail instances and data transfer OUT from a Lightsail instance when using the instance’s private IP address are free beyond your data transfer allowance. For Lightsail CDN distributions, all data transfer out of your distribution will incur a charge once you exceed your distribution data transfer allowance.