Can a bond have a value above the nominal in the moment of maturity?
For example, can a German Government Bond (nominal of 100%) that matures this month have a value of 120% for ask value? Or will it adjust to the nominal value always?
A bond that matures in one month and pays a coupon higher than the 1-month interest rate will be priced very near than the par (value) nominal. The intrinsic value of the bond will be higher because of the upcoming coupon payment, but you'll have to compensate the seller for the lost accrued interest, which will make up for most of the price difference.
Suppose I offered you a $100 bond that matured in one month and paid a 20% coupon (not annualized). So in one month I'm going to pay you the $100 bond value plus a $20 coupon. What would you pay for that bond? Pretty darn close to $120 (ignoring any credit or counterparty risk) However, I 'm going to require that you pay me for the interest that has accrued to that point, so the price of that bond will be reduced by close to $20, so the bond would sell near its par value.
The accepted answer explains nicely the concept of accrued interest. But one important thing to know is that it's sort of a convention to quote bond price without accrued interest. This is called a clean price (vs. a dirty price, which is just clean price plus accrued interest).
In other words, if you look at bid/ask of a German Bund, it is going to be a clean price and wouldn't reflect accrued interest. So the answer to your original question (if you are indeed looking at bid/ask) is "It will always adjust to the nominal value (par value)".