The common usage I have been exposed to is that punctuation should only be in a quotation if it’s contained in the original source. So what if your quotation ends in a period and your quotation is also the end of your sentence. Do you need two periods?
I asked my son to visit his grandma; however, he said, “No, I won’t go.”.
Is the above correct?
You don't ever want two periods in the same sentence, so just keep it to I asked my son to visit his grandma; however, he said, “No, I won’t go.” What comes after the semicolon, however, he said, is not a complete sentence and does not warrant a period anyhow. Here's how the Chicago Manual of Style puts it:
Don’t ever put two periods in a row. Put one period at the end of a declarative sentence, even if it ends with an abbreviation or a URL.
This is a question of style, and styles have changed over time. At one time it was fairly common to use two periods in a situation such as the example above, but people found that ugly. Then there was a period of time (I'm thinking ca 1960) when the period would never fall before the closing quote, even if the quote was a "complete sentence".
Current style, when a quote comes at the end of the containing sentence, is to place the period before the closing quote if the quoted text is a complete sentence, and place it after the quote otherwise. However, note that if the following sentence begins, eg, with an acronym or proper name then using this policy could result in ambiguity for the reader, and hence the second period might be justified. So to some extent the choice is at the writer's discretion.
