[verb (plain)] + ような気がする [i-adj] + ような気がする [na-adj] + な + ような気がする [noun] + の + ような気がする
ような気がする means "I have a feeling that", "it feels like": you sense something is so but can't back it up. It follows a verb or an い-adjective in the plain form, past and negative included.
A な-adjective keeps な and a noun takes の, as they do before any ような.
These go beyond the basic ような気がする.
気がする can follow the plain form directly, with the same meaning. This is common in conversation.
ような気がしない is "it doesn't feel like". In the question form, it asks whether the listener has the same feeling.