VGA to HDMI or Displayport

You can find numerous ones on Amazon, such as this:
https://www.amazon.com/GANA-Converter-Adapter-Computer-Projector/dp/B01N53IESF/ref=sr_1_8?ie=UTF8&qid=1503948204&sr=8-8&keywords=vga+to+hdmi

When looking for them, make sure it is for VGA on the computer and HDMI on the monitor and not the other way around.

The other option (my preference) is to go with a USB-HDMI adapter such as this:
https://www.amazon.com/RayCue-Adapter-Converter-Display-Graphic/dp/B06XDM64JM/ref=sr_1_3?s=electronics&ie=UTF8&qid=1503948303&sr=1-3&keywords=usb+to+hdmi

I'm assuming that this is for basic business use and not gaming.

imo, it would be better to purchase inexpensive video cards that support HDMI, rather than purchase adapters that don't have much use going forward.  Low-end Radeon cards can be had for under $20 each.

VGA is an analog standard.  Its resolutions are restricted by the crosstalk in the cables and at its best it can't compare with digital quality from HDMI or DisplayPort.  Further, VGA can't come anywhere near getting even half the performance out of that 4K monitor; it is like buying a Ferrari and then driving it in the bus lane.

I would go more for something like a cable with the adapter in it.

https://www.amazon.com/dp/B01LAYVT0O/ref=psdc_14570285011_t3_B01N53IESF
however the previous poster is correct except the one he posted first would be the wrong direction as it would need a separate cable to convert the VGA port. (the one he posted has a female VGA instead of Male VGA that is needed to connect to PC)

USB to any sort of video isn't an adapter, it's a full blown USB powered graphics device. For a permanent solution, I wouldn't go that way. Trying to adapt from VGA up won't look so good, either. Ever watched an old SD TV show on a big screen? It'll be like that.
I second getting a new, inexpensive display card. Then you'll be able to take full advantage of the bigger screen's capability.

I defy you to actually see the difference between a VGA image on a computer monitor and an hdmi or dvi-d signal, if both are running the same 1920 x 1080 resolution.   You are not getting an SD 320 image from an old TV.    The monitors I use for testing have all 3 inputs, because I work on many computers and switch between those inputs all the time.  They all look just fine.    

That said, I have to agree with Dr Khaln,  if you have a slot, a nice $35 to $45 video card makes the most sense going forward.      

this looks like a nice deal:  http://www.microcenter.com/product/481747/GeForce_GT_710_2GB
Its got 2gb memory, DirectX 12, OpenGL 4.5, HDCP support and 3D ready with dual link DVI-D and HDMI ports.    

Some others:  http://www.microcenter.com/endeca/CompareV2.aspx?returnUrl=L3NlYXJjaC9zZWFyY2hfcmVzdWx0cy5hc3B4P049NDI5NDk2NjkzNyA0Mjk0OTE1NjU3JnBhZ2U9MQ%3D%3D

My original response made an inappropriate assumption that this had to be an external solution.  I wholeheartedly agree that an inexpensive internal video card would be the most sensible solution, as long as installation isn't an issue.

The motherboards I need to expand are GA-M68MT-S2 with -

1 x PCI Express x16 slot, running at x16
2 x PCI Express x1 slots
1 x PCI slot

I assume that those cards are not proprietary for Dell PCs?

They'll work in any PC with an available PCI Express x16 slot.

Which is the version of your motherboard?

Version 1.3 or 3.1?

If it is only 1.3, buy a new computer is a better option as its PCI-e X16 is only version 2.0 which means you will only have 1080p resolution on 4K monitor.