RAW, no. RAI, definitely not.
The first option of the spell makes mention of changing only the flow within the cube. Now, if you wanted to make the water flow up, you could, but the moment the water exceeds the cube, it falls back down. This spell does not lock water into place for you to move around. It only changes the direction it flows. When a fluid encounters an obstacle, it flows around it.
Now, imagine something on top of the water, say a piece of ice. If you make the water flow upward, it will flow up, lifting the ice, and then deflecting around it to fall back down. You can only have two going at a time, so you won't be able to keep the water in the column that is holding up the ice.
But what if you got some friends? What if you controlled upward flow and your four buddies hopped on the ice with you and are shaping the water to keep it inside your lifting column? Unfortunately, there you run into the problem that the flow is weak. Now, it was probably weak enough to not lift one person, but five? I would rule a solid no at my table.
But let's say you allowed it since technically, it's not directly causing damage. You still run into the fact that all you've done is levitate. You can't actually move laterally. The water would flow faster than it would pull the ice along, meaning it would drop you out of the sky as soon as it moved one square to the side, to say nothing of the fact that lateral travel would mean you'd need more people to keep your vessel from constantly shedding water which means more weight which means more force the water can apparently exert.
Then there's RAI. Fly is a level 3 spell for a reason. At level 1, players are usually fighting a lot of melee only enemies. Flying invalidates that, to the point that some DMs ban Aarakocra players from low level games.