declara una variable a nivel formulario para llevar la cuenta:
dim account as integer
...
Private Sub screenshot_Click_1(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles screenshot.Click
'Captura la ventana activa
account = account + 1
....
oImgObj.Save("c:\Galaxy Pictures\Img" & cstr(account) & ".jpeg", System.Drawing.Imaging.ImageFormat.Jpeg)
....
se hiran guardando
img1.jpeg
img2.jpeg
img3.jpeg
...
PD:
me confundi crei que era VB
noce si te sirve el codigo