rotation - Convert quaternion to a different coordinate system

OSVR has a right-handed system: x is right, y is up, z is near.

I need to convert orientation data from a sensor in OSVR to a different right-handed system in which x is forward, y is left, and z is up.

I need to calculate the transformation that will convert a quaternion from one system to the other.

I naively tried:

void osvrPoseCallback(const geometry_msgs::PoseStampedConstPtr &msg) { // osvr to ros
    geometry_msgs::PoseStamped outputMsg;

    outputMsg.header = msg->header;
    outputMsg.pose.orientation.x = msg->pose.orientation.y;
    outputMsg.pose.orientation.y = msg->pose.orientation.z;

    outputMsg.pose.orientation.z = msg->pose.orientation.x;
    outputMsg.pose.orientation.w = msg->pose.orientation.w;

    osvrPosePub.publish(outputMsg);
    ros::spinOnce();
}

But that made things really weird. Facing north pitch is pitch, facing west, pitch is yaw, facing south, pitch is negative pitch...

How can I convert my OSVR quaternion to the a corresponding quaternion in the new coordinate system?