To suspend a dispatch queue, it is simply queue.suspend() (dispatch_suspend(queue) in Objective-C). That doesn't affect any tasks currently running, but merely prevents new tasks from starting on that queue. Also, you obviously only suspend queues that you created (not global queues, not main queue).
To resume a dispatch queue, it is queue.resume() (or dispatch_resume(queue) in Objective-C). There's no concept of “auto resume”, so you'd just have to manually resume it when appropriate.
To pass a dispatch queue around, you simply pass the DispatchQueue object that you created (or the dispatch_queue_t object that you created when you called dispatch_queue_create() in Objective-C).
In terms of canceling tasks queued on dispatch queues, this is a was introduced in iOS 8. One can item.cancel() a DispatchWorkItem (dispatch_block_cancel(block) a dispatch_block_t object in Objective-C). This cancels queued blocks/items that have not started, but does not stop ones that are underway. If you want to be able to interrupt a dispatched block/item, you have to periodically examine item.isCancelled (or dispatch_block_testcancel() in Objective-C).
See https://stackoverflow.com/a/38372384/1271826 for examples on canceling dispatch work items.
If you want to cancel tasks, you might also consider using operation queues, i.e. OperationQueue (NSOperationQueue in Objective-C). Its cancelable operations have been around for a while and you're likely to find lots of examples online. It also supports constraining the degree of concurrency with maxConcurrentOperationCount (whereas with dispatch queues you can only choose between serial and concurrent, and controlling concurrency more than that requires a tiny bit of effort on your part).
If using operation queues, you suspend and resume by changing the suspended property of the queue. And to pass it around, you just pass the NSOperationQueue object you instantiated.
Having said all of that, I'd suggest you expand your question to elaborate what sort of tasks are running in the background and articulate why you want to suspend them. There might be better approaches than suspending the background queue.
In your comments, you mention that you were using NSTimer, a.k.a. Timer in Swift. If you want to stop a timer, call timer.invalidate() to stop it. Create a new NSTimer when you want to start it again.
Or if the timer is really running on a background thread, GCD “dispatch source timers” do this far more gracefully. With a GCD timer, you can suspend/resume it just like you suspend/resume a queue, just using the timer object instead of the queue object.
