This probably refers to "running an errand" which is spoken often in English. An "errand" an be any task that must be done in another place such as going to the bank, the grocery store, etc. A "quick run" just means that the task can be completed in a short amount of time. Example: I have to make a quick run to the bank to get some money. I have to make a quick run to the store to get some milk. It might also refer to running for exercise but probably not because it would be more common to say "go for a quick run" if that was the case.

Escape! If you need to make a quick run for it, you probably did something bad and need to quickly escape or run away. "He stole your wallet so he made a quick run for it." Edit: I'm so embarrassed about my answer. Johnny is correct.

“I am going to run a quick errand” becomes “I am going to make a quick run”. Often used when we are going to drive the car to the grocery store, for example, to buy just 1 or 2 items. “I’m going to make a quick run to get some milk...I’ll be right back”. The saying implies that we won’t be gone very long.