This is an interesting question. Manfred Knodt wrote that Nicholas stayed at Friedberg in 1891 as a guest at manoeuvres. [M. Knodt, "Ernst Ludwig", pp. 139-140] Unfortunately, he did not provide a date for this visit. Knodt's biography, the on line "Amtspresse Preussens", Nicholas' letters to his mother, Alix's letters to Ernst Ludwig, and those to her friend Toni do not confirm that such a visit took place.
If this visit did take place, it must have been after Nicholas' return from his world trip. On 1/13 August, Nicholas' mother wrote to him that she envied Grand Duke Sergei for being the first to welcome Nicholas back and that she herself expected/ hoped to see Nicholas in two days, after an 'interminable and painful separation'. A visit to Darmstadt or Friedberg would not have taken place before the second half of August.
I have no access to Nicholas' full 1891 diary, but based on the information I found in papers, I doubt Nicholas travelled to Friedberg in the late summer or autumn of 1891:
"The Times" of 21 and 24 August reported that the tsar, tsarina, Nicholas, and Xenia had arrived in St. Petersburg from Peterhof and had left for Copenhagen on board the new Polar Star on 22 August.
In the second half of August, Alix was in Darmstadt. She left Darmstadt with her father at the end of August to see the manoevres at Romrod - no mention made of Friedberg.
"The Times" of 7 September 1891: the tsar and tsarevich and others had a hunting party at the island of Hveen on Saturday 5 September.
"The Times" of 12 September 1891: "DENMARK. COPENHAGEN, SEPT. 11. The army manoeuvres will take place during the coming fortnight on the island of Fuenen. One of the two divisions will be armed with magazine rifles, the other with Remingtons. The transport by railway and steamer of tho 15,000 men of all arms has begun. It is anticipated that the Czar and Cesarewitch will accompany the Danish King and Crown Prince to the manoeuvres. - A Correspondent."
The "Amtspresse Preussens" mentioned 'Kaisertage' in Kassel around 11-13 September 1891, and then Kaisermanoeuvres near Muehlhausen and Schlotheim (Thomas-Muntzerstadt, Thueringen) later in September of that year. The latter manoeuvres were too far to the east to be the manoeuvres meant by Knodt. No mention was made of Nicholas visiting Hesse.
"The Times" and the "Amtspresse Preussens" reported that Alexander III, his wife and Nicholas and Xenia were itill in Copenhagen when Grand Duke Paul's wife died on 24 September 1891. They passed through Berlin on their way back to Moscow to attend the funeral. After the funeral, the tsar and tsarina, with their family, left St Peterburg on board the Polar Star to return to Copenhagen ["The Times" of 5 October 1891]. They stayed there for about a month: the tsar and his family intended to leave Denmark on 29 October and to travel to Livadia with the Danish royal family for the tsar's silver wedding anniversary [the "Amtspresse Preussens" of 23 October 1891]. No mention was made of Nicholas stopping at Friedberg or Darmstadt.
"The Times" of 21 Ocotber: "BERLIN, OCT. 20. It now seems certain that the Czar will visit Berlin on his way to Livadia. As is usual, a considerable amount of mystery surrounds the details of the journey, contradictory reports being received from St. Petersburg and Copen hagen. A telegram from the latter city states that his Majesty, accompanied by the Czarina, tbe Cesarewitch, the Grand Duchess Xenia, the King and Queen of Denmark, and the Princess of Wales, will leave there on the 29th inst. on board the lmperial yacht Polar Star for Dantsic, going thence by rail to Berlin. In spite, how ever, of the semi-official source of this informa tion, I have good reason to believe that it is not exact, and that the Czar will arrive in Berlin on Thursday afternoon next. He will reside during his stay at the Russian Embassy. Regarding the duration of his stay nothing is yet known, but it is not expected that it will be a long one. - Our Own Correspondent. "
"The Times" of 17 November: "Prince Damrong arrived yesterday at Yalta, South-West Russia. He was greeted on landing by the Cesarewitch, who accompanied him to Livadia, where he was received by the Czar and the Russian Grand Dukes. "
"The Times" of 7 December: "ST. PETERSBURG, Dec. 6. Yesterday's official Gazette publishes the Imperial ukase appointing the expected committee to deal with the effects of the famine, under the presidency of the Heir Apparent. ... The rescript concludes with an expression of the Czar's conviction that his son will find strength in his deep love for the people and the country to fulfil the public obligations thus laid upon him. ..."
"The Times" of 21 December 1901: "The "Petit Parisien" publishes intelligence from St. Petersburg stating that the Cesarewitch, being detained in the capital by his duties as president of the Famine Relief Committee, will not be able to visit France next month, and will not now come to Paris until February next. The journal adds that his Imperial Highness will remain in the French capital about ten days, and will then proceed to London to be present at the marriage of the Duke of Clarence and Avondale. "