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No it is not for a couple of reasons. You would need a low profile card. Also the PSU in the 3440 is 260W maximum and the RTX 3070 requires a minimum of a 650W PSU.

Would there be an alternative Graphics card?
I'm working with the software ToonBoom Harmony and they recommend the NVIDIA Geforce GTX  series graphics card With the 1060 recommended and the 3070 as best. 

I'm not opposed to upgrading the PSU either.

Currently I'm meeting my needs but prepping to upgrade sometime in 2023

I do not think you can find any NVidia 3xxx 2xxx generation Low-Profile cards, there are some on 10th and 16th generation: 1010, 1030, 1050, 1630, 1650 (no-LP for 1060 or 1660)
Plus some professional (Quadro) line: P400/P620/T400/T600/T1000/A2000

I wouldn't consider 10th generation since there's no any performance advantage over P620,
P620 would be also roughly the same as T400.

As already mentioned by @JOcean - only low profile cards will fit in 3440, also GPU should come without any additional power connectors as all goes via PCIe and that slot has industry-standard limit of 75W, but 260W PSU imposes a limit of 190W (12V 16A) for a whole system (except CPU as it has separate 12V 16.5A rail via two connectors which at peaks can drain close to 200W or even 300W for Xeon W-1290 depending on PSU) and after excluding all other components (16W RAM, 8W PCH, 10W Fans, 15W USB, 40W storage, 10W other - all depending on components installed/connected and workload) it leaves max say 60W for discrete GPU.

Most of later generation low-profile GeForce cards are coming in either Dual-Slot configurations or rare single-slot versions have high heatsink or fan suction in a very unlucky position for Precision 3440 as CPU-bound PCIe x16 blue slot is next to PSU and either airflow inflow will be almost blocked or card won't fit (PSU sizing/position prevents it).
You can still install GPU into x4 PCIe slot (PCH-bound) but it will have a dual negative performance impact: less lanes and data will have to travel from CPU to PCH via DMI lanes competing with data from other sources (LAN/WiFi, Storage, USB, Audio, etc.) - although with low-end cards it doesn't really matter as it doesn't need that x16 (16GB/s) bandwidth in majority of cases anyways (and PCIe x4 with 4GB/s is enough)

So depending what PC configuration you have and how you use it these NVidia cards might be a good candidates for your upgrade:
NVidia GeForce 1650 (x4 PCIe Slot)
NVidia Quadro T600
NVidia Quadro T1000
NVidia RTX A2000 (x4 PCIe Slot) - not worth it, too expensive (£200 second-hand) and huge limitations over x4 PCIe Gen3 PCH-bound slot.

1650 and A2000 have propensity to get too hot in SFF case, high-temp air build-up disrupts CPU operations too.

I would go with any of those depending on budget:
NVidia Quadro T600
NVidia Quadro T1000
Although for some strange reasons on e-bay they are priced almost the same (sometimes T1000 even cheaper)

@sam55todd​ 

I know I’m replying 2 years later, but now that gigabyte has released a low profile 3050 that has max wattage of 70w, would you say that it would work in the precision 3440? And also, isn’t it bad for the power supply to run near max capacity? Like it will for the A2000 or the 3050 if they are under max load

One user in this forum runs the RTX 3050 on a 180w PSU.  That's though I'd recommend higher watt PSU.  The worst that will happen with a Dell PSU is it will trip if it can't handle it.

There is also a new RTX A2000E Ada 16GB vRAM, VCNRTX2000ADAS-LLP, that requires only 50W (like the RTX A1000, which however is from Ampère generation). I am considering it for my Precision 3431 when it will be available in Europe. In your opinion would this card a feasible option? And the RTX A1000 Ampère?

First, Dell validated cards for the 3431.

Second, both cards are only 50w and should work with either PSU the 3431 comes with, either the 200w or 260w, 260w preferred.

Third, check/Google the dimensions and check that against actual space inside the 3431.

Fourth, it doesn't hurt to check bottleneck against your CPU.  Not sure what you have for a CPU, but for example, Google "i5-9500 RTX A2000E Ada bottleneck."  Then choose the option that's "PC Builds."  It's a way to make sure CPU won't bottleneck GPU, at least not significantly.

Fifth, userbenchmark Precision 3431 for user builds showing what GPU's are being ran in the 3431.  Keep in mind userbenchmark doesn't mention PSU's in case anyone is using a higher watt card.

@Sgmkha14​ 
IMV operating near higher bound of PSU Wattage limit shouldn't be that bad, it would be a problem if is continuously operating above upper designed limit (not talking about infrequent reasonable spikes since it has spare deviation levels). At least that's what certification label should mean, it's by design.
Moreover nature of computing workload is highly unlikely to drain max power on all rails at the same time - (CPU + GPU + other subsystems). Probably another risk is about consistently overloading single rail (max Amps are shown separately for each one on PSU spec. label, but I guess there should be some auto-protection anyways leading to system operating instability), as well as ensuring quality connectivity in sockets.
* All just educated guesses, not my area of deep expertise.