If you have $\operatorname{MD5}(m_1||m_2)$ and you know $m_2$, and want to find $m'$ such that $\operatorname{MD5}(m'||m_2)$ reaches a known, predetermined value, then you must break MD5's preimage resistance, or at least second-preimage resistance.
This would be true even if you knew $m_1$: if you know $m_1$, then the problem becomes: find $m'$ such that $\operatorname{MD5}(m'||x) = y$ for two given values of $x$ and $y$ ($x = m_1$ and $y = \operatorname{MD5}(m'||m_2))$ in your presentation). Since $y$ is the result of MD5 on a known entry, this problem is a second-preimage attack with an extra constraint on the result. MD5 is still quite robust against second-preimage attacks, so this cannot be done with existing knowledge and technology.
Not knowing $m_1$ cannot help: if the problem is already hard with knowledge of $m_1$, then it cannot be easier without knowledge of $m_1$.
The explanation above is the general case. In some cases it can be quite easy: for instance, if $m_2$ happens to be equal to $z||m_1$ for some value $z$, then $m' = m_1||z$ would be a solution.
If you have $\operatorname{MD5}(m_1||m_2)$ and you know $m_2$, then you can trivially compute $\operatorname{MD5}(m_1||m')$ for any $m'$ which begins with $m_2$, followed by the MD padding, followed by arbitrary contents. This requires knowledge of the length of $m_1$ (but not its contents). For details, look up the length extension attack.
