A photodiode is operated in the reverse biase region since an external reverse bias voltage is applied to it and if light falls on it there is an increase in the reverse bias current by many orders.
A solar cell on the other hand works by producing a voltage across its terminals and this voltage is positive on the p type region thus its like a forward bias voltage, but the current still is the reverse bias current, i.e the current is still flowing from n to p(think of how battery works). This is just like a small battery. So a solar cell actually operates in the IV quadrant.