You mentioned your cousin's laptop, so if you have access to it, you could plug your grandparents monitor into the laptop to see if it works. Hopefully the laptop will have the same type of video connector as the desktop so you can test the cable as well.
The troublesome thing is that you said you no longer get the beep when you turn the desktop computer on. This beep is generated by the motherboard BIOS "Power on Self Test" (POST). You should normally hear one beep when you turn the computer on. During the POST, if the a problem is detected, you would hear multiple beeps.
The motherboard BIOS (BIOS is "Basic Input/Output System") is a small program stored on a chip on the motherboard. The BIOS storage is "Permanent" memory, but it can be reloaded or updated by "Flashing" it.
During the POST, the BIOS tests various system components and peripherals, like keyboard, video, hard-drive (and floppy-drive), and other various parts. It also does a test of the RAM memory by writing to the RAM and reading it back to make sure it works.
Even if the monitor (or keyboard for example) is disconnected, or if there were a hard-drive or memory failure, you should still hear 2 or more beeps. Since you don't hear any beeps, it sounds like the motherboard BIOS is not able to complete the POST, or the POST is not running at all, which points to a motherboard failure.
The other (and actually more likely) possibility is the power supply. The power supply provides various controlled voltages for the motherboard, drives, etc. Various system components require different or multiple voltages to run. The power supply provides: +5Volts, -5V, +12V, -12V, and +3.3V. The hard-drives, floppy-drives, and CD-Rom-drives will use +5V and +12V. The CPU and RAM memory will use all or nearly all of these voltages.
It could be that the voltage that powers the fan(s) is functioning, but one, or more than one of the other power supply voltages has failed. The fan(s) will run, but the motherboard will not function and will not be able to perform the POST, so no beeps.
The power supply is rated by watts. It provides various levels of current for each of the voltages and these current*voltage levels determine the total watts. Faster computers and advanced video/graphics require more power from the power supply, so power supplies are available in various wattage levels.
A power supply for a "modern" desktop computer would likely be in the range of 350 Watts, to 650 Watts. It could be more, 850 Watts or 1000 Watts or more for a "Monster overclocked super gaming system". The average power supply provided with a stock name brand computer (like yours, HP) would probably be rated at 450 Watts or less.
The power supply is fairly easy to replace.
- Make sure to buy a power supply with a wattage rating at least the same as the one you are replacing, or preferably a higher rating to prevent future failures. If the one you are replacing is 350 Watts, I would find one that is 450 Watts to 500 Watts minimum.
- Unplug the power cord from the computer and remove the side cover panel.
- Take a few pictures of the location of the power supply mounting, and how and where the power supply cables are connected to the motherboard, drives, fans, etc.
- On the back of the computer, locate the 4 screws that mount the power supply to the case, and remove them.
- As you remove each of the power supply cables/connectors, make note of where each came from and the orientation (position/direction) of the connector on the motherboard/drive/... Again, take plenty of pictures.
- If you have never done this before, it is a good idea to label each cable/connector with where it was remove from, as you remove it, and label each cable/connector on the new power supply to make attaching the new power supply easier.
- After the new power supply cables/connectors are attached, mount it to the computer case as the old one was, and replace the computer side panel and power cord.
- No step 8, you are done.
If you have problems or concerns along the way, post pictures (links) here and ask away.
If all this seems too much for you to attempt, try to find a small computer shop in your area, hopefully one you have used before or recommended by someone you know.
They will be able to test your existing power supply and provide the replacement if necessary. They may even install it for free if you purchase the new power supply from them. Find out the cost ahead of time and decide if you want to have it done.