No, that's not true, but there are understandable reasons why your grandmother might believe it. There are really two mistaken ideas in play here: that the East Germans did not want a reunification (they did), and that they only wanted the right to travel (they wanted much more than that). In fact, many people in East Germany wanted a reunification (Wiedervereinigung) in 1989-90. And the denial or hindrance of the right to travel was galling for many citizens of the GDR, to be sure, but it was also not the only reason for political protest, nor was it the only reason to favor reunification.
In the years after 1990, it has come to seem inevitable that the collapse of the socialist dictatorship in East Germany would lead to reunification. The fall of the Berlin Wall in November 1989 is often regarded as the beginning of a chain of events culminating in the October 1990 reunification, rather than as the end of a story of protest and state decay that began much earlier. Of course, it was both. Reunification was never inevitable; East German protesters had long lists of demands that never mentioned or even implicitly demanded reunification. And even today some former East German dissidents look back on 1990 as a missed opportunity for reform - "socialism with a human face" that might have been carried off most effectively in an independent East Germany. It is perhaps as a reaction to this misinterpretation of German history that one might believe East Germans to have been wholly uninterested in reunification. But that belief, too, is a misinterpretation. As is often the case, the truth lies somewhere in the wide middle.
The Germans named the process by which the East German government fell and the two German states were unified die Wende ("the turning point"). The beginning of die Wende is often placed at the fall of the Berlin Wall. So the story of the end of the Cold War in Germany begins with East German politician Günter Schabowski's press conference "gaffe" on November 9, 1989, announcing the immediate implementation of new travel regulations that allowed East Germans to transit the Berlin Wall freely (this led to a flood of East Germans attempting to visit West Berlin). The story then proceeds apace for about a year: the Berlin Wall is rendered superfluous and dismantled; the Socialist Unity Party (SED) gives up its monopoly on political power in December; in the same month, the hated Stasi secret police dissolves; on March 18, 1990, the only free elections in GDR history lead to a coalition government of non-communist parties with super-majority support in the Volkskammer; in May 1990 the two countries effect an economic merger; on September 20, 1990, the Volkskammer votes to ratify the so-called "Two Plus Four" unification treaty; and on October 3, 1990, the Wiedervereinigung is fully effected and the East German government ceased to exist.
There's nothing factually incorrect about this timeline, but by beginning with the fall of the Berlin Wall, we lose the context of what was happening in East German politics prior to November 9, 1989, which serves to obscure the motives of East German protesters and dissidents. In fact, the spring through the autumn of 1989 was a tumultuous time in the politics of the one-party state, both within and especially outside of the apparatus of the SED, as the Eastern Bloc governments disintegrated. Beginning in April, the Hungarian government began to disassemble the electrified fence on the border with Austria. Thousands of East Germans, already freely able to travel to Hungary, transited into the west. In July, Gorbachev publicly abandoned the Brezhnev Doctrine, the Soviet Union's post-1968 "Monroe Doctrine" equivalent, with which the USSR claimed the right to intervene in the politics of other Eastern European states. The West German embassy in Prague was crowded with thousands of East German asylum-seekers. Coalescing around pillars of nascent East German protest like the Evangelical Church and the environmental movement, in the late summer and early autumn a number of new political protest organizations (e.g., New Forum [Neues Forum], Democratic Uprising [Demokratischer Aufbruch], and the renewed and independent Social Democratic Party) were formed. Such organizations were not novel - the East German Initiative for Peace and Human Rights (Initiative für Frieden und Menschenrechte) had existed since early 1986, but this quick proliferation was a sign that matters had come to a boiling point by October.
The protesters were anything if quiet, so there's no shortage of evidence of what they believed and demanded at the time. The evidence shows that they were by no means a "reunification movement." The "Awakening '89" (Aufbruch 89) declaration, signed in September of 1989 at the founding of the New Forum, is illustrative. The declaration articulated a vision of reform and humanitarian socialism: "We want leeway for economic initiative but no degeneration into a dog-eat-dog society," it announced. "We want free, self-confident people who nevertheless act community-conscious[ly]. We want to be protected against violence and thereby not put up with a state of bailiffs and police spies." Notably, nowhere in the declaration is German reunification listed as a demand. It is not explicitly rejected, either. It simply isn't mentioned. A massive protest took place on the Alexanderplatz in Berlin on November 4. At this protest, one of many that occurred in major East German cities throughout the autumn of 1989, hundreds of thousands of East Germans gathered to listen to dissidents, reformers, and some rather anxious, defensive officials from the SED government (including, most incredibly, secretive Stasi spymaster Markus Wolf). There, again, reunification was not a theme. Placards with slogans like "Die Macht geht vom Volke aus" ("Power comes from the people") and "Sozialismus ja, Egon nein" ("Yes to Socialism, no to Egon") were hostile to the SED government, but they were reformist, not reunificationist, and often decidedly anti-capitalist. Ingo Schulze, an East German dissident, put it this way (link in German; this is my own translation): "'Visa-free all the way to Shanghai!' From the start, it was about the whole world! And it was about authorization of the New Forum and new parties, about access to the media, about free elections and above all about democratizing our own world. The call 'we are the people!' was the decisive watchword. It was actually about taking our country into our own hands. In factories, schools, universities, theaters, and institutes, those who held the trust of the majority had begun to be elected to leadership positions. That was the real revolution. Who could stop us? With every new day, the the realization of a 'socialism with a human face' seemed ever more inevitable."
Given this background, the results of the vote in East Germany's March 1990 elections are surprising. The center-right Christian Democrats - running in explicit association with their West German counterparts, and benefiting from West German Chancellor Helmut Kohl personally stumping in East Germany in advance of the election - took more than 40% of the vote on a platform advocating a quick reunification of East and West Germany. No party other than the SPD (21.9%) polled even half that percentage. Aside from the SED itself (re-branded as the "Party of Democratic Socialism" [Partei des Demokratischen Sozialismus]) (16.4%), no East German party without a western equivalent polled anywhere close to 10% of the vote. It was a stunning demonstration that, for the vast majority of East Germans, reform meant not simply an end to the SED dictatorship, but a drawing together of the two German states along western lines. From Wolfgang Grof's history of the East German Social Democrats in 1989-90 (PDF link in German; my translation): "With the commitment of the majority of the population to German unity, the reform ideas of the GDR oppositions groups, which were never homogeneous, became obsolete. ... Though the West German parties didn't stand in the election, the voters in their majority went for those parties in the GDR whose West German partners governed in the Federal Republic. Expectations of a quick and fundamental change in the living conditions in the GDR were chiefly focused to these parties. The result of the Volkskammer election was a clear vote for the entry of the GDR into the Federal Republic under Article 23 of the Federal Republic's Basic Law." This meant that the Federal Republic's constitution would not be renegotiated; the GDR would simply slip the Federal Republic's existing structure as a new set of Federal states. Liberal capitalism would be the path forward, not "socialism with a human face." Ingo Schulze again: "If you pass by the German Historical Museum today, you'll see there a copy of a very late placard shows that outlines of the Federal Republic and the GDR, together with the slogan: 'We are one people.'" The difference between the slogans at the Alexanderplatz ("we are the people") and later ("we are one people") tells the whole story.
(Continued below due to the comment character limit.)