In the ListPlot it looks like you have data points falling along several curves. What you really have is a flat list of data points and what looks like curves are not made up of sequential points, but are quite interlaced together. Have a look at the same ListPlot with the option Joined -> True

ListPlot[lstQ, ImageSize -> {1200, 339}, AspectRatio -> Full , Joined -> True, PlotRange -> All]

The data list can be split up this way into four separate runs of data. The resulting split is not perfect.

ListPlot[Transpose[Partition[lstQ, 4, 4]], ImageSize -> {1200, 339}, AspectRatio -> Full , Joined -> True, PlotRange -> All]

I think that the fitting functions will not be happy, till the data for a single curve can be isolated out. Then you can fit the three or four separately.