No, a urinary tract infection should not lead to runny stool. What it sounds like you are describing is colitis or inflammation of the large bowel. The signs can be very intermittent and can vary in severity, from normal stools to mushy to very soft, all even within one day at times. Seeing mucus or blood is not unusual as well. There is a sense of urgency to move their bowels and many of the pets will strain when having a BM, which to the owner can almost look like they are constipated. Many pets with colitis have GI discomfort and some will even vomit.
Colitis can have many causes including intestinal parasites, dietary indiscretion, food allergies or intolerances, metabolic disturbances from internal organ malfunctions, viruses, toxins, cancer or even stress. If it continues, your veterinarian will want to know the history of her signs and will need to perform a physical exam to help rule out the causes mentioned. They may perhaps suggest a change in diet or medications. Make sure they also perform a rectal exam to evaluate her anal glands. A stool sample analysis is a very important part of getting to the bottom of this as intestinal parasites are one of the most common causes.
Until you can get in contact with your vet, I would fast Venus to give her GI system a rest. If she is an adult dog, a 12-24 hr fast will not hurt her and will let her gut calm down. Continue to offer small amounts of water at a time. When you do begin feeding her again, you will want to offer frequent, small meals of something very digestible for her. These would include items like boiled hamburger or chicken and rice or pasta. No sauces or gravies, but a little non greasy broth would be OK. If the stools improve, I would continue that for a few days and then slowly wean her back onto her normal food and feeding schedule.
When they have been going on chronically, these GI disturbances can be frustrating to diagnose and treat and sometimes we don't even find the cause, but concentrate on controlling the symptoms so our pets are comfortable. In her case, since it appears to be an acute flare up, there is no reason to think that it won't resolve completely. Make sure to get in touch with your vet if she is not improving so they can get her treated appropriately and verify that there is nothing else going on to be causing these signs. Please let me know if you have any further questions.