You can attack the tendons with low weight heavy sets. Start slowly and really work the tendons into their natural power. Avoid hack squat machines, squat machines or anything that keeps you upper body into an unatural position. To explain this when you squat your body moves (actually the bar) in an 's' pattern. This 's' pattern is completely natural allowing your balancing muscles and tendons to do their job. By using a machine to maintain a strict movement you are

a) removing the need for balancing muscles
b) moving the bar in an unatural pattern causing stress in either the back or the knee or both.

That said Regular squats and leg presses are probably your best bet. While this may sound strange this is the best way to strengthen the tendons and balancing muscles in the leg. Keep the reps low (3 to 5) and work your way to a heavy weight. eg if you leg press 500pnd for 10 try increasing that to 600pnds and do ONLY 5. This should be easy. Find you range and strenght and most of all do the exercises slowly and maintain strict form. Next the the shoulder the knee is very easy to damage. Ensure that you do not go below 90 degrees with your knees as this puts unatural stress on the knees. When you knees are stronger than you can worry about deep squats or deep leg presses. If you don't know what these exercises are go to a local gym and check with a trainer for good form. Each have variations but if you keep good form and you feet at shoulder width ONLY you will be fine (imo deep squats, sumo squats and sissy squats are a very advanced movement and should be avoided as they put undo pressure on knees/back).

Also avoid leg extensions. They place the knee in a very strained position. The pull your lower leg past the 90 degree point which is the safe point for the knee. Stick to the compound movements to strengthen the whole knee and leg and you will find good results.

Also ensure you check with a doctor to make sure you knee is ok. There could be damage and left untended the cartlidge could build up causing more problems in the future. (this happened to my shoulder. I am way to stubborn for my own good).

Now when cycling don't mash the gears every ride. You can go just as fast uphill by switching to an easier gear. Essentially spinning when you can. It is the low rpm that causes problems with joints. This will save any problem with your knees etc...That said it really can't be avoided. If you do any trail riding like myself sometimes you come to technical parts where you have to mash the gears or walk. PErsonally I mash the gears

This is of course assuming you have no seat height problem. I assume not because this only occurs during your hardtraining day.

Cheers and best of luck.