I have just bought the intel X540T2 shown below and when I opened my machine I am not sure if I can plug it in any of the available PCIe slots shown below? If anyone could please advise
Technically yes, but you will not be able to reach 10 Gbps speed with them. The only slot that's fast enough is already occupied by your GPU.
You have these slots, from top to bottom:
I'm guessing a PCI-Express x16 slot – can't see it through the GPU, but most motherboards usually put an x16 slot there. It would accept your network card if you removed the GPU.
A PCI-Express x4 slot that's also occupied by the GPU, if squircle's findings are correct. It could accept your network card, probably.
A PCI-Express x1 slot (blue) with a cutout to accept any-size PCIe card. It will accept your network card, but will only provide 1 data lane. If your motherboard provides PCIe version 2.0, then 1 lane will only be enough for 5 Gbps (probably slightly below that).
(Actually, I don't know whether your card will work at all in a x1 slot – as far as I know, generally they're supposed to, but it's possible that it might deliberately refuse to.)
Another PCI-Express x1 slot (blue). Same as #2 above.
A Conventional PCI slot (white). It is not a PCI-Express slot and does not accept any PCIe cards. (And even if it did, Conventional PCI only barely provides enough speed for 1 Gbps, let alone 10 Gbps, so don't go looking for adapters.)
Another Conventional PCI slot (white). Same as #4 above – not PCIe.
Third Conventional PCI slot (white). Same as #4 above.