The short answer is no, the word “summer” is a generic noun which does not require capitalization as per rules of capitalization. This rule applies to the names of all the seasons. You do not capitalize seasons when used in writing.
This rule has an exception when the word is used in a title, is used as a proper noun, or is the first word of a sentence. When you start a sentence with “summer” or are using it in a title, it should be capitalized. If you use summer as a proper noun, such as in “Summer Semester,” it must also be capitalized.
Another exception in which you should capitalize summer is when it is personified in poetry. Apart from these exceptions, there is no need to capitalize seasons, including summer, for any writing purposes.
So, the next time you are confused about capitalization rules for seasons, read these rules carefully to avoid writing summer capitalized in the wrong places.