I can't connect my second monitor (LG UltraFine 27UL500-W) in 4k 60 hz to my MacBook M1 Pro 13inch

Grant Bennet-Alder wrote:

<< The maximum available setting is 30Hz. >>

that suggest your adapter or cables or BOTH are limited to HDMI 1.4 speeds.

Or, if the connection uses HDMI 2.0, that the use of 10-bit-per-channel color is driving a refresh rate of 30 Hz.

DisplayPort connection proves a much simpler connection. The main limitation is that the cables must be no longer than one meter.

The title of your post says that you have a "MacBook M1 Pro 13inch," but there never was a 13" MacBook Pro that had a M3 Pro chip. This raises the question of which MacBook Pro you actually have.

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14" MacBook Pros with M-series chips can have either a M1 Pro chip or a M1 Max chip. Those with a M1 Pro chip support two external displays.

MacBook Pro (14-inch, 2021) - Technical Specifications - Apple Support

The 13" MacBook Pro has a plain M1 chip. It supports one external display. You can easily tell it apart from the 14" models because it has only two USB-C (Thunderbolt) ports and does not have a MagSafe 3 port, a HDMI port, or a SDXC card slot.

MacBook Pro (13-inch, M1, 2020) - Technical Specifications - Apple Support

The LG UltraFine 27UL500-W is a UHD 4K monitor with a 60 Hz refresh rate, 10-bit-per-channel color, and three video inputs: a DisplayPort and two HDMI ports.

If I remember correctly, the bandwidth of HDMI v2.0 is not sufficient to simultaneously allow the use of all of the following:

• 4K resolution
• 10-bit-per-channel color
• 60 Hz refresh rate
• RGB 4:4:4 color representation/sampling

Something has to give. You can lower the bit depth to 8 bits per color, or lower the refresh rate to 30 Hz, or lower the color representation to RGB 4:2:2 or RGB 4:2:1. I believe that the latter change forces adjacent pixels to have the same color - effectively lowering the resolution, which is probably not what you want for computer use.

So if you are using a HDMI-to-HDMI cable (14" M4 Pro MBP), or are using a USB-C to HDMI adapter (13" or 14" MBP), there might be some HDMI v2.0 port in the path (on the Mac, on the adapter, on the monitor) that forces some tradeoff like this. You might want to check and see if the monitor menus let you select 8-bit-per-channel color as that tradeoff.

A USB-C to DisplayPort adapter might not run into this issue.

Simultaneously supports full native resolution on the built-in display at millions of colors and:

• ONE external display with up to 6K resolution at 60Hz

Thunderbolt 3 digital video output

• Native DisplayPort output over USB-C

• VGA, HDMI, DVI, and Thunderbolt 2 output supported using adapters (sold separately)

HDMI was invented for HD TV sets. it works great at its original resolution of 720i or 720p. At higher resolutions, it quickly develops issues that are complex to solve, and the cables and adapters required to solve are NOT intuitive. 

<< The maximum available setting is 30Hz. >>

that suggest your adapter or cables or BOTH are limited to HDMI 1.4 speeds.

if you want HDMI at 60 Hz, you need an adapter that is rated for HMDI 2.0 speeds, and a Certified cable labeled:

"Premium High Speed HDMI cable" or that + "with Ethernet"  --OR--

"Ultra High Speed HDMI cable" or that + "48G"

DisplayPort connection proves a much simpler connection. The main limitation is that the cables must be no longer than one meter.

Direct ThunderBolt connection, if possible, requires an Apple cable not longer than 0.8 meters, or other brands no longer than 0.5 meters.

what model-year MacBook Pro 13-in M1 PRO?

That's subtle -- but thank you for making that more correct.

It adds additional weight to the argument that using DisplayPort is simpler.