Someone who takes inspirational quotes seriously
Try credulous. Someone who is credulous will believe anything anyone wants him to believe.
I'd say such a person is susceptible or impressionable.
susceptible
Likely or liable to be influenced or harmed by a particular thing:
'patients with liver disease may be susceptible to infection'
impressionable
Easily influenced:
'a girl of eighteen is highly impressionable'
No direct connotation with quotes, though.
References:
http://www.oxforddictionaries.com/definition/english/susceptible
http://www.oxforddictionaries.com/definition/english/impressionable