I believe that, in most situations, the determining factor on getting satisfactory viewing of high definition ("hq" in your parlance) programs via a Slingbox is the upload speeds provided by your Internet Service Provider at the physical location. The transfer speeds of your wireless router -- even an older model -- are probably much higher than the upload transfer speeds of your cable modem up to the Internet. You can check these speeds by connecting a computer to your wireless router at the physical location of your Slingbox and visit a speed testing website, such as www.bandwidthplace.com or www.speakeasy.net/speedtest.
If you are NOT getting upload speeds of at least 4 megabits per second, you will likely not have satisfactory streaming of high definition video by your Slingbox. If that is the case, I would suggest contacting your ISP to see what they can offer in terms of a contract service for faster upload speeds. You may already be contracted for such high speeds, but have other problems, such as a damaged cable, that are limiting your upload speeds below the advertised for your service contract.