Theres a filter called woocommerce_available_payment_gateways:
add_filter('woocommerce_available_payment_gateways','filter_gateways',1);
function filter_gateways($gateways){
global $woocommerce;
//Remove a specific payment option
unset($gateways['paypal']);
return $gateways;
}
I’m not sure where and how you get / store the Request for Quote option, but you can access the value inside the filter_gateways function and you can remove a specific gateway with a conditional logic.