What is the direction of the sideslip induced by roll on a conventional aircraft?
To answer what you wrote: $c_{n_q}$ is the yawing moment induced by a pitching motion. In a conventional, symmetric configuration this is zero for the airframe and has a small value for propeller aircraft, depending on the rotation direction of the propeller. If the tractor propeller is spinning clockwise (when seen by the pilot), a positive pitch motion $q$ (nose up) will cause a small negative sideslip angle $\beta$. A positive pitch acceleration will add a precession moment which similarly pulls the aircraft's nose to the right. But I think you want to know something else.

To answer what you meant: $c_{n_p}$ is the yawing moment induced by a rolling motion. The axis system you propose (with z pointing up) is used to measure relative positions within the airframe, but for flight mechanics we prefer to put the x-axis in flight direction, y to the right and z down. A positive rolling motion $p = \omega_x\cdot\frac{b}{2\cdot v_{\infty}}$ (or $p = \omega_x\cdot\frac{b}{v_{\infty}}$, both definitions are used) is right-wing-down, and a positive yawing moment means the aircraft's nose moves to the right, causing negative side slip angles.

When a positive rolling motion happens in still air, the resulting yawing moment is negative, pulling the aircraft's nose left, as Federico said. Thus, $c_{n_p}$ is negative.

But the reason is different: When the pilot deflects the ailerons and causes a difference in lift on the left and right wings, and consequently a difference in induced drag, the yawing moment caused by that difference in induced drag is called $c_{n_\xi}$, the yawing moment due to aileron deflection. $c_{n_\xi}$ is positive.

Once the aircraft is in a steady roll, the roll damping $c_{l_p}$ will fully compensate the differential lift caused by aileron deflection. Now the rolling motion will increase the local angle of attack on the downward moving wing the farther you are away from the center, and decrease it on the upward moving wing. Since lift is perpendicular to local airflow, the air force vector on the downward moving wing is pointing slightly forward, and that on the upward moving wing slightly backwards. Please note that the total lift produced by each wing is roughly the same! The rolling moment of both is exactly zero once the roll rate is steady. However, the aileron deflection will decrease lift on the downward moving wing such that it just compensates for the increased angle of attack due to the rolling motion, and the same holds for the upward moving wing with its trailing-edge-down aileron deflection.

What is different is not the magnitude, but the direction of lift on both wings. This pulls the downward-moving wing forward and the upward moving wing back, causing a negative yawing moment in a positive rolling motion. That is why your conventional $c_{n_p}$ is negative.