Generally, no, you would not be charged interest. The reason for this is the concepts of the billing period and the grace period.
The billing period is a month-long period that your monthly statement covers. For example, my credit card billing period ends on the 16th of each month. So my last statement covered all transactions between May 17 and June 16.
The grace period is the amount of time between the billing period end date (also called statement date) and the due date. On my credit card, the grace period listed in the terms is at least 25 days. In actuality, my due date is on the 13th of each month, which gives me a grace period of about 27 days (more or less, depending on how many days are in the month). The payment for my most recent statement is due on July 13. If I pay my entire statement balance by that due date (and I always do), I won't pay any interest.
If I make a credit card purchase on July 12, the day before my payment due date, that purchase will be on next month's statement, which covers the billing period June 17-July 16. The payment for that purchase will be due on August 13.