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I'm just getting into PCB design and I generally have trouble learning how to use GUIs with menus and mouse clicking. Do some engineers use something more like code to generate PCBs?

Altium has a scripting language, several different languages, in fact. In some cases it may make sense to use algorithms, for example to create repetitive layouts or parts placed precisely in certain positions.

For example, I've seen it used to place parts (LEDs) in a circular pattern, though with their introduction of polar coordinate snap grids that's much less likely to be worth the hassles.

We've used algorithms to create shapes (think antennas and that kind of thing) directly in .dxf format which can then be imported into a copper layer.

In general, code is poorly suited for PCB layout purposes, particularly so for the routing task.

In EAGLE CAD, everything you can do with a mouse you can do from the command line. It also has a user language that you can essentially program in.

I wouldn't recommend it, but if you really desire you can write PCB artwork yourself, without any PCB program. The code looks like this:

G90*

1 G70*

2 G54D10*

3 G01X0Y0D02*

4 X450Y330D01*

5 X455Y300D03*

6 G54D11*

7 Y250D03*

8 Y200D03*

9 Y150D03*

10 X0Y0D02*

11 M02*

12

The line numbers at far right are not part of the file. Examining this
file without any prior knowledge of Gerber one would correctly deduce
that each line represents a particular machine command and that the
asterisk (*) is the end of command character. There seems to be
different kinds of commands: instructions beginning with G, D, M and
x,y coordinate data.

Source: https://www.artwork.com/gerber/appl2.htm

Here is a spec that shows the different commands

I use code to generate the netlist for my PCBs. (It's far, far quicker to write a for-loop to build an N-bit multiplexer than to laboriously draw all that stuff in a schematic.) Fortunately, KiCAD netlist files are just text, in a format that's undocumented but reasonably easy to reverse-engineer.

I wrote a small C# library that lets me type in what I want connected to what (in terms of reusable parametric blocks of circuitry), and it automatically spits out a netlist. Now I can just import that straight into KiCAD and start building the PCB, without having to waste an hour or so uselessly drawing a schematic for it. (The library even does some very basic checks to ensure my instructions aren't completely bogus, although I suspect KiCAD itself would do that job better.)

KiCAD PCB files, on the other hand, are also text, but appear to be far too complex to generate programmatically. Which is a shame, because KiCAD defaults to dumping all the components directly on top of each other, requiring me to spend 20+ minutes tediously separating them out again so I can see what the hell I'm doing. (KiCAD has a nasty habit of trying to move the text rather than the component it's attached to, presumably just to make PCBs harder to design?)

Perhaps some day I'll manage to also automate the initial component layout; I suspect, as others have said, that the actual "PCB design" bit will always involve the GUI though.