Indeed.
That “trigger” would be induced by the low level of inflammation always present in the gut, caused by the bacteria living there. The immune system is always fighting those bacteria, because if they escape the gut, they are deadly. The immune system fights by releasing cytokines and chemokines and by producing various T cells.
Any antibodies induced by the bacteria would not be effective against influenza, because antibodies are highly specific to a pathogen. However, the low level of inflammation always present because of the gut flora could induce the production of naive B cells, which are ready to respond to infection.
What makes you think this?