No, Gen 4 games don't commit anything to the save file about Pokemon traded away (except the full struct for the Pokemon deposited to the GTS, if any, and assuming you count that as a traded Pokemon to begin with).
However, assuming you beat the Pokemon League with some of the Pokemon you want to retrieve, you may still be in luck. Gen 4 does store non-trivial information about each entrant to the Hall of Fame, which you can easily verify with your own game. The data stored for each entrant includes:
- its species, level, nickname, gender, shininess, and original trainer's name, all in plain sight;
- its moves (plain sight in HG/SS, visible in D/P/Pt if you press A);
- its PID (what?), as the game needs that to determine the Pokemon's gender; and
- its original trainer's TID/SID, as this (plus the PID) is required for the game to know if the Pokemon is shiny (as it wants to show a shiny sprite if so).
Even better is you also get the Pokemon's ability, nature, size, and Pokeathlon stats, because those are determined by its PID.
What's missing is anything not listed above that isn't determined by the PID. This includes data like EVs, friendship, encounter details, Poke Ball, ribbons, held item, and bonus PP. You will also technically miss IVs, but if you know a bit about how the game's RNG works (introduction to the subject), knowledge of the PID can help you find what the IVs legally could have been, except in the case of egg Pokemon.
If this is enough incentive to see you trek down the path of recovering your Pokemon (and making up some data to fill in the gaps), then the rough process for achieving that would be:
- Dumping your save file. The easiest way to do this is with a hacked 3DS running a save file manager, which will write to an SD card.
- Using this tool to load your save file's Hall of Fame data and extract the partial Pokemon data files. (Disclaimer: I have not used this tool.)
- Opening your save file in PKHeX, importing the Pokemon data files extracted from the above, and exporting the modified save. If your Pokemon are Method 1 encounters (which includes all wild encounters and your starter), then PKHeX will know they're illegal from their IVs and encounter details. As above, you can deal with this by writing in plausible data with PKHeX, which may include looking up the Pokemon's PID in an RNG tool to find what the IVs should be. This tool may also assist.
- Use your hacked 3DS to write the modified save back onto the cartridge.
So, clearly there is a barrier to entry here and several limitations. But you can indeed do something to bring your Pokemon back, assuming you did take your starter, etc. through the Pokemon League.