Nope, the word "my" is a pronoun.
Why the word "my" is not a noun.
In English grammar "my" is a first person singular possessive pronoun that also acts as a determiner.
For example, in the sentence
My blue car is parked outside - the word "my" could replace a possessive noun phrase like "The speaker's".
It determines who is speaking, and also denotes possession.
What is a common noun?
If you're wondering what a common noun is, it's basically a non-specific place, thing or object. For example, the words
tree,
woman or
tea are all common nouns.
The other type of noun is called a
proper noun, and refers to a specific place or person. The words "Jane" or "London" are proper nouns because they refer to specific people or places.
Proper nouns always need to be capitalised, so it's fairly easy to tell them apart!