Is there anyway to increase my dedicated VRAM on HP laptop

I am planning to buy 'The Talos Principle' game. The minimum VRAM for this game is 526 MB and my laptop has 4 GB RAM, but the dedicated VRAM is only 128 MB. Is there anyway to increase my dedicated VRAM on an HP laptop? I have DirectX12.

Also, is it possible or would it help to run the game by connecting to the TV with HDMI cable?

Dedicated memory (aka VRAM), no, but...

The only way to increase that, is to replace the graphics adapter, which is typically not possible on laptops. Fortunately, you can just connect an eGPU (external GPU) instead. The caveat is that eGPUs aren't as mobile, and not all laptops are compatible to them.

According to HP:

When it comes to upgrading your gaming setup, detail is everything. Most eGPU docks require the host computer to have a Thunderbolt™ 3 port to connect to the external graphics dock. Thunderbolt 3 is a high-speed input/output port from Intel® that allows the eGPU to communicate with the laptop to provide accelerated graphics.

[...]

In addition to compatibility with your laptop’s connection, you’ll also want to consider an eGPU’s compatibility with the graphics card of your choice. Some eGPU docks have graphics cards pre-installed, while others give you the freedom to choose the right graphics card for you.

Shared GPU memory, maybe.

Windows 10 will let the GPU use roughly 50% of your RAM should the GPU not have enough VRAM. Whether you can upgrade your RAM depends on whether your notebook supports it.

However, for the GPU, RAM access is much slower than VRAM access, so your FPS will suffer a lot. This will likely make the game unplayable, no matter how fast your GPU is. With only 128 MB VRAM, it's very likely your GPU isn't fast enough to play this game anyway.

If your GPU is an IGP (integrated GPU), its VRAM will come from RAM. Your UEFI/BIOS will assign part of the RAM to be used exclusively for the IGP. If your UEFI/BIOS supports it, you can change how much RAM will be assigned to the IGP. Whether you can assign 526 MB or more is a different matter, however.

Also, is it possible or would it help to run the game by connecting to the TV with HDMI cable?

No. The screen only displays what your PC tells it to. A person without hands won't be able to play poker any better if he got an extra pair of eyes either.