Yes, should be the case:
# Setup
conv = torch.nn.Conv1d(in_channels=512, out_channels=512, kernel_size=1).double()
lin = torch.nn.Linear(512, 512).double()

# use same param values
with torch.no_grad():
    lin.weight = nn.Parameter(conv.weight.squeeze(2))
    lin.bias = nn.Parameter(conv.bias)

# forward
x = torch.randn(2, 512, 20).double()
out_conv = conv(x)

# permute for linear
x_lin = x.permute(0, 2, 1)
out_lin = lin(x_lin)

# check forward output
print(torch.allclose(out_lin.permute(0, 2, 1), out_conv))
> True

print((out_lin.permute(0, 2, 1) - out_conv).abs().max())
> tensor(1.2212e-15, dtype=torch.float64, grad_fn=<MaxBackward1>)

# check backward
out_conv.mean().backward()
out_lin.mean().backward()
print(torch.allclose(conv.weight.grad.squeeze(2), lin.weight.grad))
> True
print(torch.allclose(conv.bias.grad, lin.bias.grad))
> True