Are you newly diagnosed? Would you like to tell us a bit about how you came to be diagnosed, which insulins you are using MDI (multiple daily injections) or insulin pump and if you have the Freestyle Libre glucose monitoring system?
As regards your question, it very much depends on the coffee. A skinny latte is more likely to give you a spike than a coffee with cream because the skimmed milk contains more lactose (sugar) than cream. Caffeine is also a stimulant so If you are not used to it or you have a second cup when you normally just drink one it can trigger your liver to release some glucose into your blood stream rather than the coffee directly spiking your levels.
Having said that, most of us drink coffee in the morning and I am wondering if you are having a coffee maybe instead of or before breakfast and seeing a rise in your BG and attributing it to the coffee when it is more likely to be caused by Dawn Phenomenon (DP) or Foot on the Floor syndrome (FOTF) This is where the liver pumps out glucose into the blood stream in the absence of food, to give us energy to start our day. It happens to most people but obviously non diabetic people produce their own insulin to deal with it. I usually find that I need to inject 1.5 units of bolus insulin as soon as I wake up to cover FOTF but everyone if different. If you are not sure if it is the coffee or DP/FOTF, experiment with not having a coffee or anything else a couple or 3 times and see what happens to your levels. And if your levels do rise between waking up and say a couple of hours later then make a note of roughly how many mmols it rises and see how consistent that is and figure out how many units of insulin you would need to deal with that.
Good diabetes management is all about experimenting to see how your body works because we are all different. I tested lots before I got Libre but that has been a game changer for understanding how my body works as I can scan as often as I like (usually about 30 times a day) to see what is happening and that enables me to predict and plan an insulin strategy for different scenarios like FOTF or spot when my basal insulin dose needs adjusting or keeping my levels from dropping too low during exercise. It really is an amazing bit of kit.