Do not be confused by

nonimmigrant status and nonimmigrant visa. You can file Change of status application from H-4 to H-1 with CIS here in the US. If it is approved, your status changes to H-1B and you can stay here and work for the validity of period of H-1B classification. But you do not have an H-1B visa on your passport. So if you need to travel abroad and enter the US again, you may be sent to home country to get the H-1B visa by the US consulate if you try to get it somewhere else, such as Canada. It is not necessary the "must" case. Someone we got approval from F-1 to H-1B and got his H-1B visa in Canada last month. It really depends on how strong your H-1B is and how you prepare for the visa application in the US consulate where you go.