Do I need to prove the usage of my trademark during the validity period?

After registration, your trademark will be valid for 10 years (in most jurisdictions). During this time, there is no need to present proof of usage at any level, with only one exception being the United States.

As an owner of a registered trademark in the US, you will be required to submit a Declaration of Use, proof of use of your trademark in commerce, between 5th and 6th year after the registration date. You can do so by filing a Section 8 declaration. If the trademark is in use, you (or your legal representative) will be asked to provide a statement that the trademark is in use in commerce, a list of the goods or services in connection with which the trademark is in use, and a specimen (example of how the trademark is used). There is a filing fee for submitting the Declaration of Use.

If you are applying for a trademark in the United States on an intent-to-use basis, there's a similar procedure where you need to submit a Statement of Use instead. Statement of Use is proof of the trademark entering commercial use within Year 1 for intent-to-use applications. This is also subject to a fee.