Nicolo said, "I'll let you know if anything changes."
Placing the full stop before the closing quotation is the so-called US convention.
Nicolo said, "I'll let you know if anything changes".
Placing the full stop after the closing quotation is the so-called UK convention.
These are just conventions, not hard and fast rules. Poets and novelists have the artistic licence to deviate from the norm.
Nicolo said, "I'll let you know if anything changes.".
Using two end marks isn't very common. However, there are some people who tend to include one end mark before the quotation mark and another one after it.
Considering the aforementioned sentence, the first full stop is used to mark the end of the statement being quoted("I'll let you know if anything changes."). The second full stop suggests the end of the entire statement(Nicolo said....). Usage of two different end marks also depends on the situation and the meaning you'd like to convey.
Did she really say, "I love you"?
I heard him yell, "Do you love me?"
It is also correct to write, I heard him yell, "Do you love me?".(but it's not very common, as I mentioned before)
Source: Grammar Monster