I have a drain pipe located underneath under a bathroom sink.
Half of the sewer pipe is black ABS, while the other half of the pipe is made out of white PVC.
There are no threads on either side. We cannot use a threaded nut and slip washer.
No threaded couplings may be used, unless we maybe buy something which is threaded at both ends. Both sewer pipes have the same diameter.
Maybe a straight smooth (unthreaded) coupling would work?
You can use 'transition cement' to make a connection between PVC and ABS for Drain-Waste-Vent applications. It's not suitable for pressure applications. You haven't provided a picture so I can't be absolutely certain, but assuming the coupler fights tightly and the rest of the plumbing makes sense, transition cement will do the job.
There are couplers available, in rubber, with a clip at each end. They're designed to join two pipes of similar o.d. I say similar, as the clips can tighten down a few mm to accommodate a slight difference in o.d. between the two. What their materials are is inconsequential. They're designed to slide completely over the end of one pipe, offer the second up, then slid back over it, before tightening up the clips - usually stainless steel. It also means any replacements in the future are made simple. No adhesives of any kind are needed.
A picture of the 2 pipes you want to fit and the location are important in a proper answer to your question. As stated, cementing a connection together may not be advisable. You may need to attach a "P" trap adapter to mechanically place the drain. However we simply do not know the situation.
For a simple answer as to connection, there are "multipurpose" cements made that will bond PVC,CPVC and ABS. Cementing the 2 together with a "slip" connector is easy. Both items are readily available at bigbox and hardware stores. I have used this type of cement to bond PVC and ABS drains and vents for years with no issues.
