As such, print(m) works as long as before the loop you set each time:
θ = 0.01
ϵ = 1.0
but you probably want to use print("$m ") or println(m) to easily read it.
As said by @apo383 for the plot you need something more and different (and, being only 1 point each time, you should use scatter!). So:
plot()
while θ < 2
m0 = 0.1
while ϵ > 0.001
m = μ*tanh(J/(μ*θ)*m0 + h)
print("$m ")
ϵ = abs(m-m0)
m0 = m
end
display(scatter!([θ],[m]))
θ += dθ
end