Convert road bike mechanical disc brakes to hydraulic

You can’t just “connect” a mechanical cable to a hydraulic system. You’d have to upgrade the shifters/brake levers (“brifters”) to ones which are made for hydraulic brakes. Unfortunately brifters are the most expensive part of road groupsets, so this would be quite an expensive upgrade. The least expensive is Shimano Tiagra 4700. However it’s also 10 speed instead of your 9 speed so you’d have to upgrade the derailleurs, chain, cassette etc. as well.

There used to be “adapters” (TRP Parabox, Trickstuff Doppelmoppel) before hydraulic brifters became available. These were mounted somewhere on your handlebars, and through some pistons allowed you to “convert” from mechanical cables to hydraulic. However they are quite expensive and I’m not sure how well they worked.

What’s your issue with mechanical disc brakes? Maybe new and properly routed cables (+cable housing) is all you need.

It is possible using special convertors that have a hydraulic fluid reservoir that is controled by the cable from your brifters. Giant makes such an upgrade kit https://www.giant-bicycles.com/gb/conduct-hydraulic-disc-brake-upgrade-kit There are also brakes that have hydraylic fluid only in the caliper itself such as TRP HY/RD.

Otherwise, the same limitations I discussed at https://bicycles.stackexchange.com/a/72522/21133 apply. Shimano only.makes brifters from 10 speed above and those different cable pull ratio for shifting.

Mountain bikes are easier to convert to hydraulic as the brakes are often separate from the shifters on the handle bars. This makes upgrading brakes straightforward and reasonably cost effective as you can keep your old shifters. Most road bikes are have integrated brake/shifter units, this means upgrading the brakes also means you must upgrade the shifters which adds substantial to the cost. Furthermore hydraulics have been difficult to fit into the small space available in a road brake/shift lever so they tend to be found on higher end component lines. In your case this would mean having to upgrade to a system with a higher number of gears which means also replacing the derailleurs and potentially the chainrings/cranks depending on how high end you go.

This may seem ridiculous just to upgrade the brakes and it is, but it is also the result where development went. Road set ups used to separate the brakes and shifters, but there were serious advantages to integrating the brake and shifter. Next time you are climbing out of the saddle and seamlessly grab a different gear, long ago you would have had to sit down, swipe at a downtube shifter, then stand back up to make the same shift. Integrating the two components have made fitting hydraulics difficult due to the limited space and complexities of the mechanisms. This has delayed the entry of hydraulic braking into road cycling for quite a while, about 15-20 years behind mountain bikes.

You can find some brake only hydraulic levers (TRP) but you are left with figuring out shifting. Most road frames no longer support downtube shifters so you are not left with many options.