definitely does, a cold infection can protect the airways from a flu infection[1]. Cross reactivity of T cells very much exists. Another case is particular variants of covid-19 which is a corona virus granted partial immunity to influenza (one reason people suggested why both cold and flu seasons during covid peaks slowed considerably, apart from containment measures), and certain corona viruses that cause the common cold in turn also granted partial immunity against covid.

[1]https://news.yale.edu/2020/09/04/common-cold-combats-influen...