You should be able to specify HTTP headers as the third argument to your get() method as described here:
http://api.rubyonrails.org/classes/ActionDispatch/Integration/RequestHelpers.html#method-i-get
and here
http://api.rubyonrails.org/classes/ActionDispatch/Integration/Session.html#method-i-process
So, you can try something like this:
get '/my/path', nil, {'HTTP_ACCEPT' => "application/json"}