I tried it Vietnam 3 years ago.
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Your question suggests you want to make the caramel syrup part coffee-flavoured. I'd simply replace the water used in making that with espresso (if you can get real espresso). You don't need much so if you can't make it at home, perhaps get a takeaway from a coffee bar. If you can't get real espresso, something close like aeropress espresso or moka (stove-top espresso), in both cases using a lot of coffee to the amount of water. Instant espresso powder is a last resort, or for reinforcing the flavour if it turns out too weak.
If you want to make the custard part coffee flavoured, I've had success in the past infused hot milk with ground coffee before filtering. I was going for very strongly flavoured to make a latte buttercream, but you wouldn't need to. Instead as the milk starts to warm, stir in ground coffee, continue heating, then filter (through a coffee filter paper, which you can put in a sieve for a one-off if you don't have a filter cone) before you add it to the egg. Again, avoid instant coffee. Strength will be hard to get right. My best guess would be something like a tablespoon of ground coffee to 250ml of milk, but I'd suggest aiming for subtle rather than overpowering on the first attempt, and use a little less.
I referred to this recipe; you may need to modify my steps slightly if yours works differently.
In Vietnamese, it's called bánh flan. The coffee variation of bánh flan is called bánh flan cà phê.
To add coffee flavor, there are several ways:
You make plain black coffee with a coffee filter. Filtered coffee is cooked with sugar in a pan until it's thicker (caramelized).
You make plain black coffee with a coffee filter. Sugar is caramelized in a pan separately.
You mix coffee grounds/ instant coffee with hot water.
I believe the first or second way is more proper, and the last way is invented by home cooks for convenient purpose.